x-1-2-y-1-2-27xy-x-2-1-y-2-1-10xy-

Question Number 88811 by jagoll last updated on 13/Apr/20

{\begin{cases} {(x + 1)}^{2} {(y + 1)}^{2} = 27 x y \\ (x^{2} + 1) (y^{2} + 1) = 10 x y \end{cases}

Answered by john santu last updated on 13/Apr/20

l e t m e t r y

{(\sqrt{x} + \frac{1}{\sqrt{x}})}^{2} {(\sqrt{y} + \frac{1}{\sqrt{y}})}^{2} = 27

(x + \frac{1}{x}) (y + \frac{1}{y}) = 10

t a k e \sqrt{x} + \frac{1}{\sqrt{x}} = u, \sqrt{y} + \frac{1}{\sqrt{y}} = v

\Rightarrow u^{2} v^{2} = 27

\Rightarrow (u^{2} - 2) (v^{2} - 2) = 10

u^{2} v^{2} - 2 (u^{2} + v^{2}) = 6 \Rightarrow u^{2} + v^{2} = \frac{21}{2}

u^{2}, v^{2} a r e r o o t s o f w^{2} - \frac{21}{2} w + 27 = 0

w e g e t w_{1} = 6 \land w_{2} = \frac{9}{2}

t h e n u^{2} = 6 \land v^{2} = \frac{9}{2}

u^{2} - 2 = 4, w^{2} - 2 = \frac{5}{2}

(1) x + \frac{1}{x} = 4 \Rightarrow x = 2 \pm \sqrt{3}

(2) x + \frac{1}{x} = \frac{5}{2} \Rightarrow x = \frac{1}{2}; 2

s i n c e e q n (1) & (2) x a n d y s y m e t r i c,

t h e s o l u t i o n (x, y) i s (2 \pm \sqrt{3}, 2),

(2 \pm \sqrt{3}, \frac{1}{2}), (\frac{1}{2}, 2 \pm \sqrt{3}),

(2, 2 \pm \sqrt{3})

Commented by jagoll last updated on 13/Apr/20

w a w \dots c o o l l

Facebook Tweet Pin